pacman::p_load(sf, tidyverse, funModeling, blorr, corrplot, ggpubr, spdep, GWmodel, tmap, skimr, caret)In-Class_Ex2: Geographically Weighted Logistic Regression (GWLR) and Application
1. Overview
In this in-class exercise, we will learn the basic concepts and methods of Geographical Weighted Logistic Regression (GWLR) algorithm, namely:
Weighting functions (kernel)
Weighting schemes
Bandwidth
Interpreting and visualising
1.1 Model Variables
The dependent variable will be the water point status (i.e. functional, non-functional).
The independent variables are:
distance_to_primary_road
distance_to_secondary_road
distance_to_tertiary_road
distance_to_city
distance_to_town
water_point_population
local_population_1km
usage_capacity
is_urban
water_source_clean
Note: All variables are continuous, except the last three variables which are categorical.
1.2 Setting the scene
To build an explanatory model to discover factors affecting water point status in Osun State, Nigeria
2. Getting Started
2.1 Installing and loading R packages
In the code chunk below, we will install and launch these R packages into R environment.
2.2 Importing the analytical data
Two data sets will be used in this study:
osun_wp_sf.rds - Nigeria Level 2 administrative boundary (also known as Local Government Area) .
osun_rds - Water point geospatial data.
Using the code chunk below, we will extract the using read_rds of Base R.
Osun <- read_rds("rds/Osun.rds")
Osun_wp_sf <- read_rds("rds/Osun_wp_sf.rds")3. Exploratory Data Analysis
Using the code chunk below, we will examine the proportion of functional and non-functional water points, represented by “True” and “False” respectively.
Osun_wp_sf %>%
freq(input = "status")Warning: The `<scale>` argument of `guides()` cannot be `FALSE`. Use "none" instead as
of ggplot2 3.3.4.
ℹ The deprecated feature was likely used in the funModeling package.
Please report the issue at <https://github.com/pablo14/funModeling/issues>.
status frequency percentage cumulative_perc
1 TRUE 2642 55.5 55.5
2 FALSE 2118 44.5 100.0tmap_mode("view")tmap mode set to interactive viewingtm_shape(Osun)+
tm_polygons(alpha=0.4)+
tm_shape(Osun_wp_sf) +
tm_dots(col="status",
alpha=0.6)+
tm_view(set.zoom.limits = c(9,12))Using the code chunk below, we will generate the summary statistics will skimr() of Base R.
Osun_wp_sf %>%
skim()Warning: Couldn't find skimmers for class: sfc_POINT, sfc; No user-defined `sfl`
provided. Falling back to `character`.| Name | Piped data |
| Number of rows | 4760 |
| Number of columns | 75 |
| _______________________ | |
| Column type frequency: | |
| character | 47 |
| logical | 5 |
| numeric | 23 |
| ________________________ | |
| Group variables | None |
Variable type: character
| skim_variable | n_missing | complete_rate | min | max | empty | n_unique | whitespace |
|---|---|---|---|---|---|---|---|
| source | 0 | 1.00 | 5 | 44 | 0 | 2 | 0 |
| report_date | 0 | 1.00 | 22 | 22 | 0 | 42 | 0 |
| status_id | 0 | 1.00 | 2 | 7 | 0 | 3 | 0 |
| water_source_clean | 0 | 1.00 | 8 | 22 | 0 | 3 | 0 |
| water_source_category | 0 | 1.00 | 4 | 6 | 0 | 2 | 0 |
| water_tech_clean | 24 | 0.99 | 9 | 23 | 0 | 3 | 0 |
| water_tech_category | 24 | 0.99 | 9 | 15 | 0 | 2 | 0 |
| facility_type | 0 | 1.00 | 8 | 8 | 0 | 1 | 0 |
| clean_country_name | 0 | 1.00 | 7 | 7 | 0 | 1 | 0 |
| clean_adm1 | 0 | 1.00 | 3 | 5 | 0 | 5 | 0 |
| clean_adm2 | 0 | 1.00 | 3 | 14 | 0 | 35 | 0 |
| clean_adm3 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| clean_adm4 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| installer | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| management_clean | 1573 | 0.67 | 5 | 37 | 0 | 7 | 0 |
| status_clean | 0 | 1.00 | 9 | 32 | 0 | 7 | 0 |
| pay | 0 | 1.00 | 2 | 39 | 0 | 7 | 0 |
| fecal_coliform_presence | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| subjective_quality | 0 | 1.00 | 18 | 20 | 0 | 4 | 0 |
| activity_id | 4757 | 0.00 | 36 | 36 | 0 | 3 | 0 |
| scheme_id | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| wpdx_id | 0 | 1.00 | 12 | 12 | 0 | 4760 | 0 |
| notes | 0 | 1.00 | 2 | 96 | 0 | 3502 | 0 |
| orig_lnk | 4757 | 0.00 | 84 | 84 | 0 | 1 | 0 |
| photo_lnk | 41 | 0.99 | 84 | 84 | 0 | 4719 | 0 |
| country_id | 0 | 1.00 | 2 | 2 | 0 | 1 | 0 |
| data_lnk | 0 | 1.00 | 79 | 96 | 0 | 2 | 0 |
| water_point_history | 0 | 1.00 | 142 | 834 | 0 | 4750 | 0 |
| clean_country_id | 0 | 1.00 | 3 | 3 | 0 | 1 | 0 |
| country_name | 0 | 1.00 | 7 | 7 | 0 | 1 | 0 |
| water_source | 0 | 1.00 | 8 | 30 | 0 | 4 | 0 |
| water_tech | 0 | 1.00 | 5 | 37 | 0 | 20 | 0 |
| adm2 | 0 | 1.00 | 3 | 14 | 0 | 33 | 0 |
| adm3 | 4760 | 0.00 | NA | NA | 0 | 0 | 0 |
| management | 1573 | 0.67 | 5 | 47 | 0 | 7 | 0 |
| adm1 | 0 | 1.00 | 4 | 5 | 0 | 4 | 0 |
| New Georeferenced Column | 0 | 1.00 | 16 | 35 | 0 | 4760 | 0 |
| lat_lon_deg | 0 | 1.00 | 13 | 32 | 0 | 4760 | 0 |
| public_data_source | 0 | 1.00 | 84 | 102 | 0 | 2 | 0 |
| converted | 0 | 1.00 | 53 | 53 | 0 | 1 | 0 |
| created_timestamp | 0 | 1.00 | 22 | 22 | 0 | 2 | 0 |
| updated_timestamp | 0 | 1.00 | 22 | 22 | 0 | 2 | 0 |
| Geometry | 0 | 1.00 | 33 | 37 | 0 | 4760 | 0 |
| ADM2_EN | 0 | 1.00 | 3 | 14 | 0 | 30 | 0 |
| ADM2_PCODE | 0 | 1.00 | 8 | 8 | 0 | 30 | 0 |
| ADM1_EN | 0 | 1.00 | 4 | 4 | 0 | 1 | 0 |
| ADM1_PCODE | 0 | 1.00 | 5 | 5 | 0 | 1 | 0 |
Variable type: logical
| skim_variable | n_missing | complete_rate | mean | count |
|---|---|---|---|---|
| rehab_year | 4760 | 0 | NaN | : |
| rehabilitator | 4760 | 0 | NaN | : |
| is_urban | 0 | 1 | 0.39 | FAL: 2884, TRU: 1876 |
| latest_record | 0 | 1 | 1.00 | TRU: 4760 |
| status | 0 | 1 | 0.56 | TRU: 2642, FAL: 2118 |
Variable type: numeric
| skim_variable | n_missing | complete_rate | mean | sd | p0 | p25 | p50 | p75 | p100 | hist |
|---|---|---|---|---|---|---|---|---|---|---|
| row_id | 0 | 1.00 | 68550.48 | 10216.94 | 49601.00 | 66874.75 | 68244.50 | 69562.25 | 471319.00 | ▇▁▁▁▁ |
| lat_deg | 0 | 1.00 | 7.68 | 0.22 | 7.06 | 7.51 | 7.71 | 7.88 | 8.06 | ▁▂▇▇▇ |
| lon_deg | 0 | 1.00 | 4.54 | 0.21 | 4.08 | 4.36 | 4.56 | 4.71 | 5.06 | ▃▆▇▇▂ |
| install_year | 1144 | 0.76 | 2008.63 | 6.04 | 1917.00 | 2006.00 | 2010.00 | 2013.00 | 2015.00 | ▁▁▁▁▇ |
| fecal_coliform_value | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| distance_to_primary_road | 0 | 1.00 | 5021.53 | 5648.34 | 0.01 | 719.36 | 2972.78 | 7314.73 | 26909.86 | ▇▂▁▁▁ |
| distance_to_secondary_road | 0 | 1.00 | 3750.47 | 3938.63 | 0.15 | 460.90 | 2554.25 | 5791.94 | 19559.48 | ▇▃▁▁▁ |
| distance_to_tertiary_road | 0 | 1.00 | 1259.28 | 1680.04 | 0.02 | 121.25 | 521.77 | 1834.42 | 10966.27 | ▇▂▁▁▁ |
| distance_to_city | 0 | 1.00 | 16663.99 | 10960.82 | 53.05 | 7930.75 | 15030.41 | 24255.75 | 47934.34 | ▇▇▆▃▁ |
| distance_to_town | 0 | 1.00 | 16726.59 | 12452.65 | 30.00 | 6876.92 | 12204.53 | 27739.46 | 44020.64 | ▇▅▃▃▂ |
| rehab_priority | 2654 | 0.44 | 489.33 | 1658.81 | 0.00 | 7.00 | 91.50 | 376.25 | 29697.00 | ▇▁▁▁▁ |
| water_point_population | 4 | 1.00 | 513.58 | 1458.92 | 0.00 | 14.00 | 119.00 | 433.25 | 29697.00 | ▇▁▁▁▁ |
| local_population_1km | 4 | 1.00 | 2727.16 | 4189.46 | 0.00 | 176.00 | 1032.00 | 3717.00 | 36118.00 | ▇▁▁▁▁ |
| crucialness_score | 798 | 0.83 | 0.26 | 0.28 | 0.00 | 0.07 | 0.15 | 0.35 | 1.00 | ▇▃▁▁▁ |
| pressure_score | 798 | 0.83 | 1.46 | 4.16 | 0.00 | 0.12 | 0.41 | 1.24 | 93.69 | ▇▁▁▁▁ |
| usage_capacity | 0 | 1.00 | 560.74 | 338.46 | 300.00 | 300.00 | 300.00 | 1000.00 | 1000.00 | ▇▁▁▁▅ |
| days_since_report | 0 | 1.00 | 2692.69 | 41.92 | 1483.00 | 2688.00 | 2693.00 | 2700.00 | 4645.00 | ▁▇▁▁▁ |
| staleness_score | 0 | 1.00 | 42.80 | 0.58 | 23.13 | 42.70 | 42.79 | 42.86 | 62.66 | ▁▁▇▁▁ |
| location_id | 0 | 1.00 | 235865.49 | 6657.60 | 23741.00 | 230638.75 | 236199.50 | 240061.25 | 267454.00 | ▁▁▁▁▇ |
| cluster_size | 0 | 1.00 | 1.05 | 0.25 | 1.00 | 1.00 | 1.00 | 1.00 | 4.00 | ▇▁▁▁▁ |
| lat_deg_original | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| lon_deg_original | 4760 | 0.00 | NaN | NA | NA | NA | NA | NA | NA | |
| count | 0 | 1.00 | 1.00 | 0.00 | 1.00 | 1.00 | 1.00 | 1.00 | 1.00 | ▁▁▇▁▁ |
Osun_wp_sf_clean <- Osun_wp_sf %>%
filter_at(vars(status,
distance_to_primary_road,
distance_to_secondary_road,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
usage_capacity,
water_source_clean),
all_vars(!is.na(.))) %>%
mutate(usage_capacity = as.factor(usage_capacity))Things to learn from the code chunk above,
4. Correlation Analysis
In the code chunks below,
Osun_wp <- Osun_wp_sf_clean %>%
select(c(7,35:39, 42,43,46,47,57)) %>%
st_set_geometry(NULL)cluster_vars.cor = cor(
Osun_wp[,2:7])
corrplot.mixed(cluster_vars.cor,
lower="ellipse",
upper = "number",
tl.pos = "lt",
diag = "l",
tl.col = "black")
From the output above, as there is no correlation factor >0.8, it is safe to assume there is no evidence of multi-colinearity between the continuous variables.
Using the code chunk below,
model <- glm(status ~ distance_to_primary_road +
distance_to_secondary_road +
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
is_urban +
usage_capacity +
water_source_clean +
water_point_population +
local_population_1km,
data = Osun_wp_sf_clean,
family = binomial(link = "logit"))Instead of using a typical R report, we will generate a report using blr() of blorr package using the code chunk below.
blr_regress(model) Model Overview
------------------------------------------------------------------------
Data Set Resp Var Obs. Df. Model Df. Residual Convergence
------------------------------------------------------------------------
data status 4756 4755 4744 TRUE
------------------------------------------------------------------------
Response Summary
--------------------------------------------------------
Outcome Frequency Outcome Frequency
--------------------------------------------------------
0 2114 1 2642
--------------------------------------------------------
Maximum Likelihood Estimates
-----------------------------------------------------------------------------------------------
Parameter DF Estimate Std. Error z value Pr(>|z|)
-----------------------------------------------------------------------------------------------
(Intercept) 1 0.3887 0.1124 3.4588 5e-04
distance_to_primary_road 1 0.0000 0.0000 -0.7153 0.4744
distance_to_secondary_road 1 0.0000 0.0000 -0.5530 0.5802
distance_to_tertiary_road 1 1e-04 0.0000 4.6708 0.0000
distance_to_city 1 0.0000 0.0000 -4.7574 0.0000
distance_to_town 1 0.0000 0.0000 -4.9170 0.0000
is_urbanTRUE 1 -0.2971 0.0819 -3.6294 3e-04
usage_capacity1000 1 -0.6230 0.0697 -8.9366 0.0000
water_source_cleanProtected Shallow Well 1 0.5040 0.0857 5.8783 0.0000
water_source_cleanProtected Spring 1 1.2882 0.4388 2.9359 0.0033
water_point_population 1 -5e-04 0.0000 -11.3686 0.0000
local_population_1km 1 3e-04 0.0000 19.2953 0.0000
-----------------------------------------------------------------------------------------------
Association of Predicted Probabilities and Observed Responses
---------------------------------------------------------------
% Concordant 0.7347 Somers' D 0.4693
% Discordant 0.2653 Gamma 0.4693
% Tied 0.0000 Tau-a 0.2318
Pairs 5585188 c 0.7347
---------------------------------------------------------------From the report above, we can exclude the distance_to_primary_road and distance_to_secondary_road as there are not statistically significant at their p-value is > 0.05.
5. Building a Logistic Regression Model
The validity of a cut-off is measured using sensitivity, specificity, and accuracy.
Sensitivity : The % of correctly classified events out of all events TP/ (TP + FN)
Specificity : The % of correctly classified non-events out of all non-events (TN / (TN + FP)
Accuracy : The % of correctly classified observations over all observations (TP + TN)/ (TP + FP + TN + FN)
blr_confusion_matrix(model, cutoff = 0.5)Confusion Matrix and Statistics
Reference
Prediction FALSE TRUE
0 1301 738
1 813 1904
Accuracy : 0.6739
No Information Rate : 0.4445
Kappa : 0.3373
McNemars's Test P-Value : 0.0602
Sensitivity : 0.7207
Specificity : 0.6154
Pos Pred Value : 0.7008
Neg Pred Value : 0.6381
Prevalence : 0.5555
Detection Rate : 0.4003
Detection Prevalence : 0.5713
Balanced Accuracy : 0.6680
Precision : 0.7008
Recall : 0.7207
'Positive' Class : 1From the output above, we can see that the model can pick up the true positive (72.1%) better than the true negative (61.5%)
6. Building Geographically Weighted Logistic Regression Model
6.1 Converting from sf to sp data frame
Using the code chunk below,
Osun_wp_sp <- Osun_wp_sf_clean %>%
select(c(status,
distance_to_tertiary_road,
distance_to_city,
distance_to_town,
water_point_population,
local_population_1km,
is_urban,
usage_capacity,
water_source_clean)) %>%
as_Spatial()
Osun_wp_spclass : SpatialPointsDataFrame
features : 4756
extent : 182502.4, 290751, 340054.1, 450905.3 (xmin, xmax, ymin, ymax)
crs : +proj=tmerc +lat_0=4 +lon_0=8.5 +k=0.99975 +x_0=670553.98 +y_0=0 +a=6378249.145 +rf=293.465 +towgs84=-92,-93,122,0,0,0,0 +units=m +no_defs
variables : 9
names : status, distance_to_tertiary_road, distance_to_city, distance_to_town, water_point_population, local_population_1km, is_urban, usage_capacity, water_source_clean
min values : 0, 0.017815121653488, 53.0461399623541, 30.0019777713073, 0, 0, 0, 1000, Borehole
max values : 1, 10966.2705628969, 47934.343603562, 44020.6393368124, 29697, 36118, 1, 300, Protected Spring 6.2 Building fixed bandwidth GWR model
6.2.1 Computing fixed bandwidth
Calulating the distance matrix
bw.fixed <- bw.ggwr(status ~
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
is_urban +
usage_capacity +
water_source_clean +
water_point_population +
local_population_1km,
data = Osun_wp_sp,
family = "binomial",
approach = "AIC",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE)Take a cup of tea and have a break, it will take a few minutes.
-----A kind suggestion from GWmodel development group
Iteration Log-Likelihood:(With bandwidth: 95768.67 )
=========================
0 -2890
1 -2837
2 -2830
3 -2829
4 -2829
5 -2829
Fixed bandwidth: 95768.67 AICc value: 5681.18
Iteration Log-Likelihood:(With bandwidth: 59200.13 )
=========================
0 -2878
1 -2820
2 -2812
3 -2810
4 -2810
5 -2810
Fixed bandwidth: 59200.13 AICc value: 5645.901
Iteration Log-Likelihood:(With bandwidth: 36599.53 )
=========================
0 -2854
1 -2790
2 -2777
3 -2774
4 -2774
5 -2774
6 -2774
Fixed bandwidth: 36599.53 AICc value: 5585.354
Iteration Log-Likelihood:(With bandwidth: 22631.59 )
=========================
0 -2810
1 -2732
2 -2711
3 -2707
4 -2707
5 -2707
6 -2707
Fixed bandwidth: 22631.59 AICc value: 5481.877
Iteration Log-Likelihood:(With bandwidth: 13998.93 )
=========================
0 -2732
1 -2635
2 -2604
3 -2597
4 -2596
5 -2596
6 -2596
Fixed bandwidth: 13998.93 AICc value: 5333.718
Iteration Log-Likelihood:(With bandwidth: 8663.649 )
=========================
0 -2624
1 -2502
2 -2459
3 -2447
4 -2446
5 -2446
6 -2446
7 -2446
Fixed bandwidth: 8663.649 AICc value: 5178.493
Iteration Log-Likelihood:(With bandwidth: 5366.266 )
=========================
0 -2478
1 -2319
2 -2250
3 -2225
4 -2219
5 -2219
6 -2220
7 -2220
8 -2220
9 -2220
Fixed bandwidth: 5366.266 AICc value: 5022.016
Iteration Log-Likelihood:(With bandwidth: 3328.371 )
=========================
0 -2222
1 -2002
2 -1894
3 -1838
4 -1818
5 -1814
6 -1814
Fixed bandwidth: 3328.371 AICc value: 4827.587
Iteration Log-Likelihood:(With bandwidth: 2068.882 )
=========================
0 -1837
1 -1528
2 -1357
3 -1261
4 -1222
5 -1222
Fixed bandwidth: 2068.882 AICc value: 4772.046
Iteration Log-Likelihood:(With bandwidth: 1290.476 )
=========================
0 -1403
1 -1016
2 -807.3
3 -680.2
4 -680.2
Fixed bandwidth: 1290.476 AICc value: 5809.719
Iteration Log-Likelihood:(With bandwidth: 2549.964 )
=========================
0 -2019
1 -1753
2 -1614
3 -1538
4 -1506
5 -1506
Fixed bandwidth: 2549.964 AICc value: 4764.056
Iteration Log-Likelihood:(With bandwidth: 2847.289 )
=========================
0 -2108
1 -1862
2 -1736
3 -1670
4 -1644
5 -1644
Fixed bandwidth: 2847.289 AICc value: 4791.834
Iteration Log-Likelihood:(With bandwidth: 2366.207 )
=========================
0 -1955
1 -1675
2 -1525
3 -1441
4 -1407
5 -1407
Fixed bandwidth: 2366.207 AICc value: 4755.524
Iteration Log-Likelihood:(With bandwidth: 2252.639 )
=========================
0 -1913
1 -1623
2 -1465
3 -1376
4 -1341
5 -1341
Fixed bandwidth: 2252.639 AICc value: 4759.188
Iteration Log-Likelihood:(With bandwidth: 2436.396 )
=========================
0 -1980
1 -1706
2 -1560
3 -1479
4 -1446
5 -1446
Fixed bandwidth: 2436.396 AICc value: 4756.675
Iteration Log-Likelihood:(With bandwidth: 2322.828 )
=========================
0 -1940
1 -1656
2 -1503
3 -1417
4 -1382
5 -1382
Fixed bandwidth: 2322.828 AICc value: 4756.471
Iteration Log-Likelihood:(With bandwidth: 2393.017 )
=========================
0 -1965
1 -1687
2 -1539
3 -1456
4 -1422
5 -1422
Fixed bandwidth: 2393.017 AICc value: 4755.57
Iteration Log-Likelihood:(With bandwidth: 2349.638 )
=========================
0 -1949
1 -1668
2 -1517
3 -1432
4 -1398
5 -1398
Fixed bandwidth: 2349.638 AICc value: 4755.753
Iteration Log-Likelihood:(With bandwidth: 2376.448 )
=========================
0 -1959
1 -1680
2 -1530
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2376.448 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2382.777 )
=========================
0 -1961
1 -1683
2 -1534
3 -1450
4 -1416
5 -1416
Fixed bandwidth: 2382.777 AICc value: 4755.491
Iteration Log-Likelihood:(With bandwidth: 2372.536 )
=========================
0 -1958
1 -1678
2 -1528
3 -1445
4 -1411
5 -1411
Fixed bandwidth: 2372.536 AICc value: 4755.488
Iteration Log-Likelihood:(With bandwidth: 2378.865 )
=========================
0 -1960
1 -1681
2 -1532
3 -1448
4 -1414
5 -1414
Fixed bandwidth: 2378.865 AICc value: 4755.481
Iteration Log-Likelihood:(With bandwidth: 2374.954 )
=========================
0 -1959
1 -1679
2 -1530
3 -1446
4 -1412
5 -1412
Fixed bandwidth: 2374.954 AICc value: 4755.482
Iteration Log-Likelihood:(With bandwidth: 2377.371 )
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2377.371 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2377.942 )
=========================
0 -1960
1 -1680
2 -1531
3 -1448
4 -1414
5 -1414
Fixed bandwidth: 2377.942 AICc value: 4755.48
Iteration Log-Likelihood:(With bandwidth: 2377.018 )
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413
Fixed bandwidth: 2377.018 AICc value: 4755.48 bw.fixed[1] 2377.371gwlr.fixed <- ggwr.basic(status ~
distance_to_tertiary_road +
distance_to_city +
distance_to_town +
is_urban +
usage_capacity +
water_source_clean +
water_point_population +
local_population_1km,
data = Osun_wp_sp,
bw = bw.fixed,
family = "binomial",
kernel = "gaussian",
adaptive = FALSE,
longlat = FALSE) Iteration Log-Likelihood
=========================
0 -1959
1 -1680
2 -1531
3 -1447
4 -1413
5 -1413 gwlr.fixed ***********************************************************************
* Package GWmodel *
***********************************************************************
Program starts at: 2022-12-17 14:46:17
Call:
ggwr.basic(formula = status ~ distance_to_tertiary_road + distance_to_city +
distance_to_town + is_urban + usage_capacity + water_source_clean +
water_point_population + local_population_1km, data = Osun_wp_sp,
bw = bw.fixed, family = "binomial", kernel = "gaussian",
adaptive = FALSE, longlat = FALSE)
Dependent (y) variable: status
Independent variables: distance_to_tertiary_road distance_to_city distance_to_town is_urban usage_capacity water_source_clean water_point_population local_population_1km
Number of data points: 4756
Used family: binomial
***********************************************************************
* Results of Generalized linear Regression *
***********************************************************************
Call:
NULL
Deviance Residuals:
Min 1Q Median 3Q Max
-129.368 -1.750 1.074 1.742 34.126
Coefficients:
Estimate Std. Error z value Pr(>|z|)
Intercept 3.540e-01 1.055e-01 3.354 0.000796
distance_to_tertiary_road 1.001e-04 2.040e-05 4.910 9.13e-07
distance_to_city -1.764e-05 3.391e-06 -5.202 1.97e-07
distance_to_town -1.544e-05 2.825e-06 -5.466 4.60e-08
is_urbanTRUE -2.667e-01 7.474e-02 -3.569 0.000358
usage_capacity1000 -6.206e-01 6.966e-02 -8.908 < 2e-16
water_source_cleanProtected Shallow Well 4.947e-01 8.496e-02 5.823 5.79e-09
water_source_cleanProtected Spring 1.279e+00 4.384e-01 2.917 0.003530
water_point_population -5.098e-04 4.476e-05 -11.390 < 2e-16
local_population_1km 3.452e-04 1.779e-05 19.407 < 2e-16
Intercept ***
distance_to_tertiary_road ***
distance_to_city ***
distance_to_town ***
is_urbanTRUE ***
usage_capacity1000 ***
water_source_cleanProtected Shallow Well ***
water_source_cleanProtected Spring **
water_point_population ***
local_population_1km ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 6534.5 on 4755 degrees of freedom
Residual deviance: 5688.9 on 4746 degrees of freedom
AIC: 5708.9
Number of Fisher Scoring iterations: 5
AICc: 5708.923
Pseudo R-square value: 0.129406
***********************************************************************
* Results of Geographically Weighted Regression *
***********************************************************************
*********************Model calibration information*********************
Kernel function: gaussian
Fixed bandwidth: 2377.371
Regression points: the same locations as observations are used.
Distance metric: A distance matrix is specified for this model calibration.
************Summary of Generalized GWR coefficient estimates:**********
Min. 1st Qu. Median
Intercept -3.7021e+02 -4.3797e+00 3.5590e+00
distance_to_tertiary_road -3.1622e-02 -4.5462e-04 9.1291e-05
distance_to_city -5.4555e-02 -6.5623e-04 -1.3507e-04
distance_to_town -8.6549e-03 -5.2754e-04 -1.6785e-04
is_urbanTRUE -7.3554e+02 -3.4675e+00 -1.6596e+00
usage_capacity1000 -5.5889e+01 -1.0347e+00 -4.1960e-01
water_source_cleanProtected.Shallow.Well -1.8842e+02 -4.7295e-01 6.2378e-01
water_source_cleanProtected.Spring -1.3630e+03 -5.3436e+00 2.7714e+00
water_point_population -2.9696e-02 -2.2705e-03 -1.2277e-03
local_population_1km -7.7730e-02 4.4281e-04 1.0548e-03
3rd Qu. Max.
Intercept 1.3755e+01 2171.6373
distance_to_tertiary_road 6.3011e-04 0.0237
distance_to_city 1.5921e-04 0.0162
distance_to_town 2.4490e-04 0.0179
is_urbanTRUE 1.0554e+00 995.1840
usage_capacity1000 3.9113e-01 9.2449
water_source_cleanProtected.Shallow.Well 1.9564e+00 66.8914
water_source_cleanProtected.Spring 7.0805e+00 208.3749
water_point_population 4.5879e-04 0.0765
local_population_1km 1.8479e-03 0.0333
************************Diagnostic information*************************
Number of data points: 4756
GW Deviance: 2815.659
AIC : 4418.776
AICc : 4744.213
Pseudo R-square value: 0.5691072
***********************************************************************
Program stops at: 2022-12-17 14:47:49 From the output above, as AIC reduced from 5708.9(non-geographical weighted regression model) to 4418.8 (geographical weighted regression model), we can determine that there was an improvement in the regression model.
6.3 Model assessment
6.3.1 Converting SDF into sf data frame
To assess the performance of the gwLR, first, we will covert the SDF object in as data frame using the code chunk below.
gwr.fixed <- as.data.frame(gwlr.fixed$SDF)Next, in the code chunks below, we will label yhat values greater or equal to 0.5 into 1, else into 0. The result of the logic comparison operation will be saved into a field called most.
gwr.fixed <- gwr.fixed %>%
mutate(most = ifelse(
gwr.fixed$yhat >= 0.5, T, F))gwr.fixed$y <- as.factor(gwr.fixed$y)
gwr.fixed$most <- as.factor(gwr.fixed$most)
CM <- confusionMatrix(data = gwr.fixed$most, reference = gwr.fixed$y)
CMConfusion Matrix and Statistics
Reference
Prediction FALSE TRUE
FALSE 1833 268
TRUE 281 2374
Accuracy : 0.8846
95% CI : (0.8751, 0.8935)
No Information Rate : 0.5555
P-Value [Acc > NIR] : <2e-16
Kappa : 0.7661
Mcnemar's Test P-Value : 0.6085
Sensitivity : 0.8671
Specificity : 0.8986
Pos Pred Value : 0.8724
Neg Pred Value : 0.8942
Prevalence : 0.4445
Detection Rate : 0.3854
Detection Prevalence : 0.4418
Balanced Accuracy : 0.8828
'Positive' Class : FALSE
From the output above, we can see improvement the accuracy (from 67.4% to 88.5%). This is due to improvements in both sensitivity (from 72.1% to 86.7%) and specificity (from 61.5% to 89.9%).
6.4 Visualising gwLR
Osun_wp_sf_selected <- Osun_wp_sf_clean %>%
select(c(ADM2_EN, ADM2_PCODE,
ADM1_EN, ADM1_PCODE,
status))gwr_sf.fixed <- cbind(Osun_wp_sf_selected, gwr.fixed)6.4.1 Visualising coefficient estimator
The code chunks below is used to create an interactive point symbol map.
tmap_mode("view")tmap mode set to interactive viewingprob_T <- tm_shape(Osun) +
tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf.fixed) +
tm_dots(col = "yhat",
border.col = "gray60",
border.lwd = 1) +
tm_view(set.zoom.limits = c(8,14))
prob_Ttertiary_TV <- tm_shape(Osun) +
tm_polygons(alpha = 0.1) +
tm_shape(gwr_sf.fixed) +
tm_dots(col = "distance_to_tertiary_road_TV",
border.col = "gray60",
border.lwd = 1) +
tm_view(set.zoom.limits = c(8,14))
tertiary_TVVariable(s) "distance_to_tertiary_road_TV" contains positive and negative values, so midpoint is set to 0. Set midpoint = NA to show the full spectrum of the color palette.7. Reference
Atkinson PM, German SE, Sear DQ and Clark MJ (2003) “Exploring the relations between riverbank erosion and geomorphological controls using geographically weighted logistic regression”. Geographical Analysis 35(1): 58–82.